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arXiv · 2301.10447

On algebraic and non-algebraic neighborhoods of rational curves

Serge Lvovski
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Abstract

We prove that for any $d>0$ there exists an embedding of the Riemann sphere $\mathbb P^1$ in a smooth complex surface, with self-intersection $d$, such that the germ of this embedding cannot be extended to an embedding in an algebraic surface but the field of germs of meromorphic functions along $C$ has transcendence degree $2$ over $\mathbb C$. We give two different constructions of such neighborhoods, either as blowdowns of a neighborhood of the smooth plane conic, or as ramified coverings of a neighborhood of a hyperplane section of a surface of minimal degree. The proofs of non-algebraicity of these neighborhoods are based on a classification, up to isomorphism, of algebraic germs of embeddings of $\mathbb P^1$, which is also obtained in the paper.

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Serge Lvovski. 2024-01-19. On algebraic and non-algebraic neighborhoods of rational curves. https://doi.org/10.17323/1609-4514-2024-24-3-391-405

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